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Mean Field Methods for Nuclear Structure

Mean Field Methods for Nuclear Structure. Nguyen Van Giai Institut de Physique Nucléaire Université Paris-Sud, Orsay. Part 1: Ground State Properties: Hartree-Fock and Hartree-Fock-Bogoliubov Approaches Part 2: Nuclear Excitations: The Random Phase Approximation. Nguyen Van Giai.

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Mean Field Methods for Nuclear Structure

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  1. Mean Field Methods for Nuclear Structure Nguyen Van Giai Institut de Physique Nucléaire Université Paris-Sud, Orsay Part 1: Ground State Properties: Hartree-Fock and Hartree-Fock-Bogoliubov Approaches Part 2: Nuclear Excitations: The Random Phase Approximation Istanbul, part I Nguyen Van Giai

  2. Outline of part 1 • - Introduction • - Non-relativistic energy density functional • - Densities and Potentials • - HF and HFB in spherical symmetry • - Illustrative examples • - Summary Istanbul, part I Nguyen Van Giai

  3. Microscopic approaches to many-body, finite nuclear systems • Theoretical models based on effective interactions between nucleons: - Nuclear shell model - Mean fieldapproaches (and beyond): -Non-Relativistic (Skyrme forces, Gogny force) -Relativistic (RMF,RHF) - Molecular dynamics • goingaway from stability regions, we need a theoretical framework which can bepredictiveand able to handle new situations (continuum, pairing correlations in continuum). • theHartree-Fock + Random Phase Approximation (and their extensions to include pairing effects) can be used from unstable nuclei to neutron star crust. Istanbul, part I Nguyen Van Giai

  4. Hartree-Fock, and HF-Bogoliubov for systems with pairing correlations Istanbul, part I Nguyen Van Giai

  5. Energy Density Functional in Hartree-Fock Istanbul, part I Nguyen Van Giai

  6. Effective Interaction: Skyrme force particle-hole channel: particle-particle channel: Skyrme interaction zero-range Pairing channel: Istanbul, part I Nguyen Van Giai

  7. Densities • Normal density, or density matrix • Abnormal density, or pairing tensor Istanbul, part I Nguyen Van Giai

  8. One-body densities in Hartree-Fock Istanbul, part I Nguyen Van Giai

  9. The Energy Density Functional Istanbul, part I Nguyen Van Giai

  10. The Skyrme-HF equations Variations with respect to single-particle wave functions: Istanbul, part I Nguyen Van Giai

  11. The Skyrme-HF effective masses Istanbul, part I Nguyen Van Giai

  12. The Skyrme-HF central potentials Istanbul, part I Nguyen Van Giai

  13. The spin-orbit and Coulomb potentials Istanbul, part I Nguyen Van Giai

  14. The center-of-mass correction Istanbul, part I Nguyen Van Giai

  15. Spherical case: radial equations in r-space Istanbul, part I Nguyen Van Giai

  16. Potentials Densities Effective masses Spin-orbit potentials From: Bender et al., Revs.Mod.Phys., 75, 121(2003) Istanbul, part I Nguyen Van Giai

  17. N-Z Binding Energy Errors A=N+Z From: Bender et al., Revs.Mod.Phys., 75, 121(2003) Istanbul, part I Nguyen Van Giai

  18. 2-neutron separation energies From: Bender et al., Revs.Mod.Phys., 75, 121(2003) Istanbul, part I Nguyen Van Giai

  19. Single-particle energies From: Bender et al., Revs.Mod.Phys., 75, 121(2003) Istanbul, part I Nguyen Van Giai

  20. r.m.s. radii From: Bender et al., Revs.Mod.Phys., 75, 121(2003) Istanbul, part I Nguyen Van Giai

  21. Generalization to Hartree-Fock-Bogoliubov Istanbul, part I Nguyen Van Giai

  22. HFB densities in spherical case • Nuclear density • Abnormal (or pairing) density • Kinetic energy density • Spin density Istanbul, part I Nguyen Van Giai

  23. The Hartree-Fock-Bogoliubov Equations Istanbul, part I Nguyen Van Giai

  24. Hartree-Fock field and pairing field Istanbul, part I Nguyen Van Giai

  25. Enuc= ESkyrme+ Epair [ r,k] Finite-TemperatureHFB DT(r) = Vpair kT(r) fi=(1+eEi/kT)-1 where : Istanbul, part I Nguyen Van Giai

  26. Quasiparticle continuum Istanbul, part I Nguyen Van Giai

  27. Treatment of quasiparticle continuum (1) Istanbul, part I Nguyen Van Giai

  28. Treatment of quasiparticle continuum (2) Istanbul, part I Nguyen Van Giai

  29. Treatment of quasiparticle continuum (3) Istanbul, part I Nguyen Van Giai

  30. Discretization by box boundary condition • Alternatively, one can enclose the system in a box of radius R. • The quasiparticle spectrum is calculated with the boundary condition that the wave function vanishes at r=R. • One thus obtains a discrete set of states forming a complete basis in the box. Istanbul, part I Nguyen Van Giai

  31. illustration: Ni isotopes Istanbul, part I Nguyen Van Giai

  32. E. Khan, N. Sandulescu Nguyen Van Giai Istanbul, part I Nguyen Van Giai

  33. Inner Crust Matter ~ r0 ~ 0.001r0 ~ 0.5 r0 Crystal lattice structures Istanbul, part I Nguyen Van Giai

  34. Elementary cells Wigner-Seitz cell Elementary cell Lattice Istanbul, part I Nguyen Van Giai

  35. Density in the Wigner-Seitz Cells N.Sandulescu,Nguyen Van Giai,R.J.Liotta, Phys.Rev.C69(2004)045802 Istanbul, part I Nguyen Van Giai

  36. Pairing Field in the Wigner-Seitz Cells N.Sandulescu, Phys.Rev.C70 (2004) 025801 Istanbul, part I Nguyen Van Giai

  37. SUMMARY • A self-consistent theory of nuclear ground states. • Pairing and continuum effects are treated. • Applications to the description of unstable nuclei. • Applications to the physics of the inner crust of neutron stars. Istanbul, part I Nguyen Van Giai

  38. Lectures on:Mean Field Methods for Nuclear StructureList of references for further reading • 1. P. Ring, P. Schuck, “The Nuclear Many-Body Problem”, Springer-Verlag (New York, 1980) • 2. Hartree-Fock calculations with Skyrme’s interaction. I: spherical nuclei, D. Vautherin, D.M. Brink, Phys. Rev. C 5, 626 (1972) • 3. Hartree-Fock calculations with Skyrme’s interaction. II: axially deformed nuclei, D. Vautherin, Phys. Rev. C 7, 296 (1973) • 4. A Skyrme parametrization from subnuclear to neutron star densities, E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer: Part I, Nucl. Phys. A 627, 710 (1997); Part II, Nucl. Phys. A 635, 231 (1998); Erratum to Part II, Nucl. Phys. A 643, 441 (1998) • 5. Self-consistent mean-field models for nuclear structure, M. Bender, P.-H. Heenen, P.-G. Reinhard, Revs. Mod. Phys. 75, 121 (2003) • 6. Hartree-Fock-Bogoliubov description of nuclei near the neutron drip line, J. Dobaczewski, H. Flocard, J. Treiner, Nucl.Phys. A 422, 103 (1984) • 7. Mean-field description of ground state properties of drip line nuclei: pairing and continuum effects, J. Dobaczewski, W. Nazarewicz, T.R. Werner, J.-F. Berger, C.R. Chinn, J. Dechargé, Phys. Rev. C 53, 2809 (1996) • 8. Pairing and continuum effects in nuclei close to the drip line, M. Grasso, N. Sandulescu, N. Van Giai, R. Liotta, Phys. Rev. C 64, 064321 (2001) • 9. Nuclear response functions, G.F. Bertsch, S.F. Tsai, Phys. Rep. 12 C (1975) • 10. A self-consistent description of the giant resonances including the particle continuum, K.F. Liu, N. Van Giai, Phys. Lett. B 65, 23 (1976) • 11. Continuum quasiparticle random phase approximation and the time-dependent HFB approach, E. Khan, N. Sandulescu, M. Grasso, N. Van Giai, Phys. Rev. C 66, 024309 (2002) • 12. Self-Consistent Description of Multipole Strength in Exotic Nuclei I: Method, J. Terasaki, J. Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, M. Stoitsov, Phys. Rev. C 71, 034310 (2005) • 13. Self-consistent description of multipole strength: systematic calculations, J. Terasaki, J. Engel, ArXiv nucl-th/0603062 Istanbul, part I

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